Christian Krattenthaler
We compute the number of perfect matchings of an $M\times N$ Aztec rectangle where $|N-M|$ vertices have been removed along a line. A particular case solves a problem posed by Propp. Our enumeration results follow from certain identities for Schur functions, which are established by the combinatorics of nonintersecting lattice paths.
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We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindström-Gessel-Viennot methodology to find the number of tilings of a quartered lozenge hexagon.
Helfgott and Gessel gave the number of domino tilings of an Aztec Rectangle with defects of size one on the boundary of one side. In this paper we extend this to the case of domino tilings of an Aztec Rectangle with defects on all boundary sides.
We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.
We are counting the lattice rectangles that can be constructed inside several planar shapes and identify the corresponding sequences in the OEIS.
We consider several new families of graphs obtained from Aztec rectangle and augmented Aztec rectangle graphs by trimming two opposite corners. We prove that the perfect matchings of these new graphs are enumerated by powers of $2$, $3$, $5$, and $11$. The result yields a proof of a conjectured posed by Ciucu. In addition, we reveal a hidden relation between our graphs and the hexagonal dungeons introduced by Blum.
We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box
We prove combinatorially that the parity of the number of domino tilings of a region is equal to the parity of the number of domino tilings of a particular subregion. Using this result we can resolve the holey square conjecture. We furthermore give combinatorial proofs of several other tiling parity results, including that the number of domino tilings of a particular family of rectangles is always odd.
The Tangent Method of Colomo and Sportiello is applied to the study of the asymptotics of domino tilings of large Aztec rectangles, with some fixed distribution of defects along a boundary. The associated Non-Intersecting Lattice Path configurations are made of Schröder paths whose weights involve two parameters $γ$ and $q$ keeping track respectively of one particular type of step and of the area below the paths. We derive the arctic curve for an arbitrary distribution of defects, and illustrate our result with a number of examples involving different classes of boundary defects.
In this paper we determine the interaction of diagonal defect clusters in regions of an Aztec rectangle that scale to arbitrary points on its symmetry axis (in earlier work we treated the case when this point was the center of the scaled Aztec rectangle). We use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary. In several of the treated situations this seems not to be accomplishable by previous methods. Our applications include the case of two long neutral strings, which …
Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex ($6$V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a $s\times n$ square lattice ($s\leq n$). In this paper, we derive the analytic expression of the arctic curve, for $a=b=1$ and $c=\sqrt{2}$ ($Δ=0$), while keeping the ratio $s/n \,\in [0,1]$ as a free parameter. The computation relies on the tangent method. We also consider domino tilings of double Aztec rectangles and show via the tangent method …
Domino Tilings of Cruciform Regions
本文推广Ciucu的铺砖公式，为十字形区域提供生成函数公式，研究其多米诺铺砖数目的计算方法。
Round Aztec windows, a dual of the Aztec diamond theorem and a curious symmetry of the correlation of diagonal slits
本文研究了Aztec矩形和环面Aztec钻石中特定孔洞形状的多米诺平铺数量，证明了其平铺数和完美匹配数可由简单乘积公式表示，并揭示了斜对角裂缝关联的对称性。
A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diab…
Propp's benzels and Lai's nearly symmetric hexagons with holes
本文提出对称图完美匹配因子分解定理的新版本，解决了Propp的4个六角格三聚体铺砌计数公开问题，得到带孔六边形菱形铺砌半因子分解结果，解决Lai的2个公开问题并推广相关结果，还为Aztec矩形完美匹配数公式提供了更简洁证明。
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